1
Gravitation as the result of the reintegration of
migrated electrons and positrons to their atomic
nuclei.
Osvaldo Domann
Germany
odomann@yahoo.com
This paper presents the mechanism of gravitation based on an approach where the energy of an
electron or positron is radially distributed in space. The energy is stored in fundamental
particles (FPs) that move radially and continuously through a focal point in space, point where
classically the whole energy of a subatomic particle is thought to be concentrated. FPs store the
energy in longitudinal and transversal rotations defining corresponding angular momenta.
Forces between subatomic particles are the product of the interactions of their FPs. The laws of
interactions between fundamental particles are postulated in that way, that the linear momenta
for all the basic laws of physics can subsequently be derived from them, linear momenta that are
generated out of opposed pairs of angular momenta of fundamental particles.
1. Introduction.
To explain the mechanism of gravitation it is
necessary first to show how the energies of
electrons and positrons (Basic Subatomic Particles
≡ BSPs) are distributed in space.
The total energy 𝐸𝑒 = √𝐸𝑜2 + 𝐸𝑝2 = 𝐸𝑠 + 𝐸𝑛 of a
BSP is distributed in space as follows:
𝑑𝐸𝑒 = 𝐸𝑒 𝑑𝜅 = 𝜈 𝐽𝑒
(1)
𝑑𝐸𝑠 = 𝐸𝑠 𝑑𝜅 = 𝜈 𝐽𝑠
𝑑𝐸𝑛 = 𝐸𝑛 𝑑𝜅 = 𝜈 𝐽𝑛
where
𝐸𝑠 =
𝐸𝑜2
√𝐸𝑜2 + 𝐸𝑝2
𝐸𝑝2
𝐸𝑛 =
𝑎𝑛𝑑
√𝐸𝑜2 + 𝐸𝑝2
1 𝑟𝑜
𝑑𝛾
𝑑𝜅 = 2 𝑑𝑟 sin 𝜑 𝑑𝜑 2 𝑟𝑟
2𝜋
FPs moving to the focal point (regenerating FPs)
have longitudinal 𝐽𝑠 and transversal 𝐽𝑛 angular
momenta and associated to them respectively a
longitudinal regenerating field 𝑑𝐻̄𝑠 defined as
𝑑𝐻̄𝑠 = 𝐻𝑠 𝑑𝜅 𝑠̄ = √𝜈 𝐽𝑠 𝑑𝜅 𝑠̄
(4)
𝑤𝑖𝑡ℎ
𝐻𝑠2 = 𝐸𝑠
and a transversal regenerating field 𝑑𝐻̄𝑛 defined
as
𝑑𝐻̄𝑛 = 𝐻𝑛 𝑑𝜅 𝑛̄ = √𝜈 𝐽𝑛 𝑑𝜅 𝑛̄
(5)
𝑤𝑖𝑡ℎ
𝐻𝑛2 = 𝐸𝑛
For the total field magnitude 𝐻𝑒 it is
𝐻𝑒2 = 𝐻𝑠2 + 𝐻𝑛2
(2)
with 𝑑𝜅 a distribution function for the energy
which is inverse proportional to the square distance
to the focal point, giving the fraction of energy in
the volume 𝑑𝑉 = 𝑑𝑟 𝑟 𝑑𝜑 𝑟 sin 𝜑 𝑑𝛾.
FPs leaving the focal point (emitted FPs) have only
longitudinal angular momenta 𝐽𝑒 and associated to
it a longitudinal emitted field 𝑑𝐻̄𝑒 defined as
𝑑𝐻̄𝑒 = 𝐻𝑒 𝑑𝜅 𝑠̄𝑒 = √𝜈 𝐽𝑒 𝑑𝜅 𝑠̄𝑒
(3)
𝑤𝑖𝑡ℎ
𝐻𝑒2 = 𝐸𝑒
Figure 1. Unit vector 𝒔̄ 𝒆 for an emitted FP and unit
vectors 𝒔̄ and 𝒏̄ for a regenerating FP of a BSP moving
with 𝒗 ≠ 𝒄
In Fig. 1 the vector 𝑠̄𝑒 is a unit vector in the
moving direction of the emitted fundamental
2
particle (FP). The vector 𝑠̄ is a unit vector in the
moving direction of the regenerating FP. The
vector 𝑛̄ is a unit vector transversal to the moving
direction of the regenerating FP and oriented
according the right screw rule relative to the
velocity 𝑣̄ of the BSP.
which are constant for non relativistic speeds.
Nucleons are composed of electrons and positrons
which are concentrated in the range of 0 ≤ 𝛾 ≤
0.1 of the curve of Fig. 3 where the attractions and
repulsions between them are zero.
Conclusion: Basic subatomic particles (BSPs) are
structured particles with longitudinal and
transversal angular momenta, with positive or
negative emitted angular momenta (charge) and
with a transversal field 𝑑𝐻̄𝑛 (mechanical
momentum and magnetic moment).
2. Gravitation.
The interaction law between FPs of static BSPs
(Coulomb) follows the cross product between
longitudinal angular momentum |𝐽̄𝑠1 × 𝐽̄𝑠2 | =
𝐽𝑠1 𝐽𝑠2 sin 𝛽 of the FPs, which is zero for the
distance 𝑑 = 0 between BSPs because of 𝛽 = 𝜋.
The differential linear momentum 𝑑𝑝 on a BSP is
generated out of pairs of opposed angular momenta
𝐽̄𝑛 at the regenerating FPs of the BSP. At a moving
BSP, opposed pairs of angular momenta are
generated at the regenerating FPs because of the
axial symmetry of the FPs relative to the velocity
of the BSP as shown in Fig. 1.
In Fig. 2 the differential linear momentum 𝑑𝑝2 at
BSP 2 is generated by pairs of opposed angular
momenta 𝐽̄𝑛2 of regenerating FPs.
Figure 3. Linear momentum
  d / ro
𝒑𝒔𝒕𝒂𝒕 as function of
between two static BSPs with equal focal radii
𝒓𝒐𝟏 = 𝒓𝒐𝟐
Electrons and positrons of a stable nucleon migrate
slowly into the range of 0.1 ≤ 𝛾 ≤ 1.8 polarizing
the nucleon, and are subsequently reintegrated with
high speed when their FPs cross with FPs of the
remaining electrons and positrons of the nucleon,
because of 𝛽 < 𝜋 as shown in Fig. 4 for Neutron
1.
Figure 2. Generation of angular momenta 𝑱𝒏 at
regenerating fundamental particles of two static basic
subatomic particles at the distance 𝒅
Fig. 3 shows the linear momentum between two
BSPs as a function of the distance 𝑑. The variable
𝑟𝑜 represents the radii of the focus of the BSPs,
Figure 4. Transmission of momentum
𝟏 to neutron 𝟐
𝒅𝒑 from neutron
3
The opposed transversal angular momenta 𝐽̄𝑛
responsible for the dH n fields at the FPs of each
reintegrating electron or positron of a nucleon, are
captured by regenerating FPs of an electron or
positron of another nucleon, generating the
attracting force between the nucleons as shown in
Fig. 4. Reintegrating BSP ’b’ of neutron 1 hands
over its opposed angular momenta to BSP ’p’ of
neutron 2 with the result, that the linear
momentum 𝑑𝑝𝑏 of BSP ’b’ reduces to zero while
the linear momentum 𝑑𝑝𝑝 of BSP ’p’ increases by
the same magnitude complying with the
conservation law for momentum.
To calculate the gravitation force induced by the
reintegration of migrated BSPs, we need to know
the number of migrated BSPs in the time Δ𝑡 for a
neutral body with mass 𝑀.
The following equation was derived in [6] for the
induced gravitation force generated by one
reintegrated electron or positron
𝐹𝑖 =
√ℎ 𝜈𝑛 √𝑚𝑝
𝑑𝑝
= ∫ ∫
𝛥𝑡
4 𝐾 𝑑 2
𝐺𝑟𝑎𝑣
𝑤𝑖𝑡ℎ
∫∫
(6)
Δ𝐺1 Δ𝐺2 = 𝐺 4 𝐾 𝑀1 𝑀2
√ℎ 𝜈𝑛 √𝑚𝑝 ∫ ∫𝐺𝑟𝑎𝑣
(10)
Because of energy exchange quantization derived
in [6] which states that energy is quantized in
energy quanta of a resting electron 𝐸𝑜 = ℎ 𝜈𝑜
with 𝜈𝑜 = 1.2373 ⋅ 1020 𝑠 −1 , we replace ℎ 𝜈𝑛
by ℎ 𝜈𝑜 in eq. (6) and get
Δ𝐺1 Δ𝐺2 = 2.1493 ⋅ 1016 𝑀1 𝑀2
(11)
Δ𝐺1 Δ𝐺2 = 𝛾𝐺2 𝑀1 𝑀2
and
Δ𝐺 = 𝛾𝐺 𝑀
𝑤𝑖𝑡ℎ
(12)
𝛾𝐺 = 1.4661 ⋅ 108 𝑘𝑔−1
Calculation example: The number of migrated
BSPs that are reintegrated at the sun and the earth
in the time Δ𝑡 are respectively, with 𝑀⊙ =
1.9891 ⋅ 1030 𝑘𝑔 and 𝑀† = 5.9736 ⋅ 1024 𝑘𝑔
Δ𝐺⊙ = 2.9162 ⋅ 1038
𝑎𝑛𝑑
(13)
32
Δ† = 8.7579 ⋅ 10
The power exchanged between two masses due to
gravitation is
𝑃𝐺 = 𝐹𝐺 𝑐
= 2.4662
𝐺𝑟𝑎𝑣
where 𝑚𝑝 is the mass of the electron and ℎ the
Planck constant and
Δ𝑡 = 𝐾 𝑟𝑜2
𝑟𝑜 = 3.8590 ⋅ 10−13 𝑚
(7)
𝐾 = 5.4274 ⋅ 104 𝑠/𝑚2
The direction of the force 𝐹𝑖 on the capturing BSP
is independent of the sign of the reintegrating BSP
and is always oriented to the reintegrating BSP.
For two bodies with masses 𝑀1 and 𝑀2 and
where the number of reintegrated BSPs in the time
Δ𝑡 is respectively Δ𝐺1 and Δ𝐺2 it must be
𝑀1 𝑀2
𝐹𝐺 = 𝐹𝑖 Δ𝐺1 Δ𝐺2 = 𝐺 (8)
𝑑 2
−11
𝐺 = 6.6726 ⋅ 10
As the direction of the force 𝐹𝑖 is the same for
reintegrating electrons Δ−𝐺 and positrons Δ+𝐺 it is
(9)
Δ𝐺 = |Δ−𝐺 | + |Δ+𝐺 |
and we get that
𝑃𝐺 = 𝑐 √ℎ 𝜈𝑜 √𝑚𝑝
4 𝐾 𝑑 2
Δ𝐺1 Δ𝐺2 ∫ ∫
(14)
𝐺𝑟𝑎𝑣
The power exchanged between the sun and the
earth is, with 𝑑⊙† = 1.49476 ⋅ 1011 𝑚
𝑀⊙ 𝑀†
𝑃𝐺 = 𝐹𝐺 𝑐 = 𝐺 2 𝑐
𝑑⊙†
(15)
𝑃𝐺 = 7.8156 ⋅ 1019 𝐽/𝑠
3. Dark matter.
In the previous sections we have seen that the
origin of the gravitation force is the induced force
due to the reintegration of migrated BSPs in the
direction of the two gravitating bodies. When a
BSP is reintegrated to a neutron, the two BSPs of
different signs that interact produce an equivalent
current in the direction of the positive BSP as
shown in Fig. 5.
4
𝑝𝑏
𝐾𝐷𝑎𝑟𝑘
=
𝑤𝑖𝑡ℎ
𝛥𝑜 𝑡
𝑑
1 ℎ
𝐾𝐷𝑎𝑟𝑘 = 2 𝛥𝑜 𝑡
= 4.09924 ⋅ 10−14 𝑁𝑚
𝑑𝐹𝑅 =
𝐾𝐷𝑎𝑟𝑘
(20)
The total force is
𝐹𝑅 =
Figure 5. Resulting current due to reintegration of
migrated BSPs
As the numbers of positive and negative BSPs that
migrate in one direction at one neutron are equal,
no average current should exists in that direction in
the time Δ𝑡. It is
(16)
Δ𝑅 = Δ+𝑅 + Δ−𝑅 = 0
We now assume, that because of the power
exchange between the two neutrons (15), a
synchronization exists between the reintegration of
BSPs of equal sign in the direction orthogonal to
the distance between the two neutrons, resulting in
parallel currents of equal signs that generate an
attracting force between the neutrons. Thus the
total attracting force between the two neutrons is
produced first by the induced force FG and second
by the parallel currents FR of reintegrating BSPs.
FT = FG + FR
with
M1 M2
FG = G (17)
d2
M1 M2
FR = R d
𝐾𝐷𝑎𝑟𝑘
𝑀1 𝑀2
Δ𝑅1 Δ𝑅2 = 𝑅 𝑑
𝑑
(21)
We get
Δ𝑅1 Δ𝑅2 =
𝑅
𝐾𝐷𝑎𝑟𝑘
𝑀1 𝑀2
(22)
or
Δ𝑅1 Δ𝑅2 = 𝛾𝑅2 𝑀1 𝑀2
𝑅
𝛾𝑅2 =
𝐾𝐷𝑎𝑟𝑘
𝑤𝑖𝑡ℎ
(23)
and
Δ𝑅 = 𝛾𝑅 𝑀
(24)
The total attracting force gives
𝐹𝑇 = 𝐹𝐺 + 𝐹𝑅 = [
𝐺 𝑅
+ ] 𝑀1 𝑀2
𝑑2 𝑑
(25)
For sub-galactic distances the induced force 𝐹𝐺 is
predominant, while for galactic distances the force
of parallel reintegrating BSPs 𝐹𝑅 predominates, as
shown in Fig. 6.
To obtain an equation for the force 𝐹𝑅 we start
with an equation that was deduced in [6] for the
linear momentum when electrons are bent through
a crystal, equation that is based on the interaction
of parallel currents.
1 5.8731 ℎ 𝜈𝑜
(18)
𝑝𝑏 = Δ𝑙 𝑛
4 64 𝑐
𝑑
with
𝜈𝑜 = 1.2373 ⋅ 10 20 𝑠 −1
Δ𝑙 = 5.2843 ⋅ 10−11 𝑚
Δ𝑜 𝑡 = 8.0821 ⋅ 10−21 𝑠
(19)
The force for one pair of parallel BSPs (𝑛 = 1) is
given by
Figure 6. Gravitation forces at sub-galactic and galactic
distances.
Calculation example:
For the sun with 𝑣𝑜𝑟𝑏 = 220 𝑘𝑚/𝑠 and 𝑀2 =
𝑀⊙ = 2 ⋅ 1030 𝑘𝑔 and a distance to the core of
the Milky Way of 𝑑 = 25 ⋅ 1019 𝑚 we get a
centrifugal force of
5
2
𝑣𝑜𝑟𝑏
𝐹𝑐 = 𝑀2 = 3.872 ⋅ 1020 𝑁
𝑑
(26)

With the mass of the core of the Milky Way of
𝑀1 = 4 ⋅ 106 𝑀⊙ and
𝑀1 𝑀2
𝐹𝑐 = 𝐹𝑇 ≈ 𝐹𝑅 = 𝑅 𝑑
𝑅 = 6.05 ⋅ 10−27 𝑁𝑚/𝑘𝑔2
(27)

and with
𝐹𝐺 = 𝐹𝑅
we get

(28)
𝑑𝑔𝑎𝑙 =
𝐺
= 1.103 ⋅ 1016 𝑚
𝑅

justifying our assumption that 𝐹𝑇 ≈ 𝐹𝑅 because
the distance between the sun and the core of the
Milky Way is 𝑑 ≫ 𝑑𝑔𝑎𝑙 .
We also have that
𝑅
𝛾𝑅 = √
= 3.842 ⋅ 10−7 𝑘𝑔−1
𝐾𝐷𝑎𝑟𝑘
(29)
If we compare with 𝛾𝐺 = 1.4661 ⋅ 108 𝑘𝑔−1 for
the induced force we see that 𝛾𝑅 is very small.
Note: The flattening of galaxies´ rotation curve
was derived based on the assumption that the
gravitation force is composed of an induced
component and a component due to parallel
currents of reintegrating BSPs and, that for galactic
distances the induced component can be neglected.
Note: We also may assume that the
synchronization of the reintegrating BSPs in the
orthogonal direction of the two neutrons results in
parallel currents of opposed signs, generating a
repulsive force between the two neutrons.
4. Summary of main
characteristics of the proposed
model.
The main characteristics of the proposed model,
with gravitation a part of it, are:
 The
approach
is
Lorentz
invariant,
quantification and probability are inherent to it.
 The energy of a BSP is stored in the
longitudinal angular momenta of emitted
fundamental particles. The rotation sense of the
longitudinal angular momentum defines the







charge of the BSP.
All the basic laws of physics (Coulomb,
Ampere, Lorentz, Maxwell, Gravitation,
bending of particles and interference of
photons,
Bragg,
Laue,
Schroedinger,
Stern-Gerlach) are mathematically derived
from the proposed model, proving that the
approach is in accordance with experimental
data.
All known forces are derived as rotors from one
vector field generated by the longitudinal and
transversal angular momenta of fundamental
particles.
The coexistence of BSPs of equal charge in
the atomic nucleus does not require the
definition of a special strong force nor
additional mediating particles (gluons).
The emission of particles from heavy atomic
nucleus does not require the definition of a
special weak force nor additional mediating
particles.
Gravitation has its origin in the reintegration of
migrated BSPs to nuclei of bodies. No special
mediating particles are required (gravitons).
The gravitation force is composed of an induced
component and a component due to parallel
currents of reintegrating BSPs. For galactic
distances the induced component can be
neglected, what explains the flattening of
galaxies´ rotation curve (no dark matter is
required).
The interacting particles for all types of
interactions (electromagnetic, strong, weak,
gravitation) are the FPs with their longitudinal
and transversal angular momenta.
The wave character of the photon is defined as a
sequence of BSPs with potential opposed
transversal linear momenta, which are generated
by transversal angular momenta of FPs that
comply with specific symmetry conditions.
Light that moves through a gravitation field can
only lose energy, what explains the red shift of
light from far galaxies (no expansion of the
universe is required).
The addition of a wave to a particle (de Broglie)
is effectively replaced by a relation between the
particles focal radius and its energy. Deflection
of particles such as the electron is now a result of
the quantified bending linear momentum
between BSPs due to the quantification of
energy exchange.
Quantum mechanics constructed with the focal
radius instead of the de Broglie wave-length
6
gives a differential equation where the wave
function is differentiated two times towards time
and one towards space. The uncertainty relations
form pairs of canonical conjugated variables
between "energy and space" and "momentum
and time”. The Schrödinger equation results as a
particular time independent case of the wave
packet.
 The new quantum mechanics theory, based on
wave functions derived with the focal radius, is
in accordance with the quantum mechanics
theory based on the correspondence principle.
 As the model relies on BSPs permitting the
transmission of linear momentum at infinite
speed via FPs, it is possible to explain that
entangled photons show no time delay when
they change their state.
 The two possible states of the electron spin are
replaced by the two types of electrons defined
by the present theory, namely the accelerating
and decelerating electrons.
 The magnetic spin moment 𝜇𝑠 that is
responsible for the splitting of the atomic beam
in the Stern-Gerlach experiment is replaced by
the quantized bending moment of parallel
currents of electrons.
5. References.
1. Albrecht Lindner. Grundkurs Theoretische
Physik. Teubner Verlag, Stuttgart 1994.
2. Benenson ⋅ Harris ⋅ Stocker ⋅ Lutz.
Handbook of Physics. Springer Verlag 2001.
3. Stephen G. Lipson. Optik. Springer Verlag
1997.
4. B.R. Martin & G. Shaw. Particle Physics.
John Wiley & Sons 2003.
5. Max
Schubert
/
Gerhard
Weber.
Quantentheorie,
Grundlagen
und
Anwendungen. Spektrum, Akad. Verlag
1993.
6. O. Domann. Definition of a space with
fundamental particles with longitudinal
and transversal rotational momentum,
and deduction of the linear momentum
which generates electrostatic, magnetic,
induction and gravitational forces. June
2003. http://www.odomann.com/